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Algorithmic Analysis of Polygonal Hybrid Systems G ERARDO S CHNEIDER V ERIMAG G RENOBLE Algorithmic Analysis of Polygonal Hybrid Systems – p.1/66

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Page 1: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Algorithmic Analysis of PolygonalHybrid Systems

GERARDO SCHNEIDER

VERIMAG

GRENOBLE

Algorithmic Analysis of Polygonal Hybrid Systems – p.1/66

Page 2: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid Systems• Hybrid Systems: interaction between discrete and

continuous behaviors• Examples: thermostat, automated highway

systems, air traffic management systems, roboticsystems, chemical plants, etc.

Model: Hybrid Automatalabel

invariant

dynamics

guard reset

x = M

x ≤ M

x = 3 − x

x ≥ m

x = −x

OffOn

x = m /γ

Algorithmic Analysis of Polygonal Hybrid Systems – p.2/66

Page 3: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid Systems

Model: Hybrid Automatalabel

invariant

dynamics

guard reset

x = M

x ≤ M

x = 3 − x

x ≥ m

x = −x

OffOn

x = m /γ

Algorithmic Analysis of Polygonal Hybrid Systems – p.2/66

Page 4: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid SystemsExample: Swimmer in a whirlpool

e10

e9

e12

e11

e2

e4

e5

e8

e1x0

e6e7

e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

Page 5: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid SystemsExample: Swimmer in a whirlpool

e10

e9

e12

e11

e2

e4

e5

e8

e1x0

e6e7

e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

Page 6: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid SystemsExample: Swimmer in a whirlpool

e10

e9

e12

e11

e2

e4

e5

e8

e1x0

e6e7

e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

Page 7: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid SystemsExample: Swimmer in a whirlpool

e10

e9

e12

e11

e2

e4

e5

e8

e1x0

e6e7

e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

Page 8: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid SystemsExample: Swimmer in a whirlpool

e10

e9

e12

e11

e2

e4

e5

e8

e1x0

e6e7

e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

Page 9: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Hybrid SystemsExample: Swimmer in a whirlpool

e10

e9

e12

e11

e2

e4

e5

e8

e1x0

e6e7

e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.3/66

Page 10: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Polygonal Differential InclusionSystems (SPDIs)

• A partition of the plane into convexpolygonal regions

• A constant differential inclusion for each region

x ∈ ∠b

aif x ∈ Ri

e3e2

e4

e5

e9

e12 e1

e8

e11

e7e6

e10

Algorithmic Analysis of Polygonal Hybrid Systems – p.4/66

Page 11: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Polygonal Differential InclusionSystems (SPDIs)

• A partition of the plane into convexpolygonal regions

• A constant differential inclusion for each region

x ∈ ∠b

aif x ∈ Ri

x′

Ri

x

ba

b

∠b

a

a

Algorithmic Analysis of Polygonal Hybrid Systems – p.4/66

Page 12: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Polygonal Differential InclusionSystems (SPDIs)

• The “swimmer” is a hybrid system• Hybrid Automata?

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

Page 13: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Polygonal Differential InclusionSystems (SPDIs)

• The “swimmer” is a hybrid system• Hybrid Automata?

e3e2

e4

e5

e9

e12 e1

e8

e11

e7e6

e10

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

Page 14: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Polygonal Differential InclusionSystems (SPDIs)

• The “swimmer” is a hybrid system• Hybrid Automata?

e2e3

e9

e12e4

e3

e1

e2

e12

e11

e1

e8

e7

e8

e11

e7e6

e10

e6

e5

e4

e5

e9

e10

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

Page 15: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Polygonal Differential InclusionSystems (SPDIs)

• The “swimmer” is a hybrid system• Hybrid Automata?

x = a7 x = a8

x = a4

Inv(`2)

x = a2

x = e3

R2

x = a1

x = e2x ∈ ∠

b

a

x = e7x = e6

x = e8

x = e1

x = e10 x = e11

x = e12

x = e9

x = e4

x = e5

Inv(`4) Inv(`3)

Inv(`1)

Inv(`8)Inv(`7)Inv(`6)

x = a6

Inv(`5)

x = a5

R1R5

R8R7R6

R3R4

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

Page 16: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Polygonal Differential InclusionSystems (SPDIs)

• The “swimmer” is a hybrid system• Hybrid Automata?

We will use the “geometric” representation instead ofthe hybrid automata

Algorithmic Analysis of Polygonal Hybrid Systems – p.5/66

Page 17: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Overview of the presentation• Motivation and Contributions• Algorithm for Reachability Problem (SPDIs)• Implementation – SPeeDI• Algorithm for Phase Portrait construction

(SPDIs)• Other 2 dim Hybrid Systems

• Between Decidability and Undecidability• Undecidability results

• Summary of Results and Perspectives

Algorithmic Analysis of Polygonal Hybrid Systems – p.6/66

Page 18: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and Contributions

Algorithmic Analysis of Polygonal Hybrid Systems – p.7/66

Page 19: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsScientific Context (Reachability)

dim

reachability

Planar 2−dim

Decidable

Undecidable

3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 20: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsScientific Context (Reachability)

dim

reachability

PCDs

Planar 2−dim

Decidable

Undecidable

3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 21: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsScientific Context (Reachability)

dim

reachability

PCDs

Planar 2−dim

Decidable

Undecidable

3−dim

Piecewise Constant Derivative Systeme3

e2

e4

e5

e9

e12 e1

e8

e11

e7e6

e10

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 22: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsScientific Context (Reachability)

dim

reachability

PCDs

Planar 2−dim

Decidable

Undecidable

3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 23: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsScientific Context (Reachability)

dim

reachability

PCDs

PCDs

Planar 2−dim

Decidable

Undecidable

3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 24: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsScientific Context (Reachability)

dim

reachability

PCDs

PCDs

Planar 2−dim

Decidable

Undecidable

3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 25: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsChallenge

dim

reachability

Nondeterministic

PCDs

Extensionsof

PCDs

PCDs

PCDs

Planar 2−dim

Decidable

Undecidable

3−dim

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 26: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsContributions (Reachability)

dim

SPDIs

PCDs

Extensionsof

Extensionsof

PCDs

PCDs

PCDs

Planar 2−dim

Decidable

Undecidable

3−dim

ProblemOpen

reachability

Algorithmic Analysis of Polygonal Hybrid Systems – p.8/66

Page 27: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and ContributionsScientific Context (Phase Portrait)

• Phase Portrait for PCDs• Numerical algorithms for computing Viability

Kernels

Algorithmic Analysis of Polygonal Hybrid Systems – p.9/66

Page 28: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Motivation and Contributions

Contributions (Phase Portrait)

• Phase Portrait for SPDIs• Viability Kernel• Controllability Kernel

Algorithmic Analysis of Polygonal Hybrid Systems – p.9/66

Page 29: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Reachability Analysis for SPDIs

Algorithmic Analysis of Polygonal Hybrid Systems – p.10/66

Page 30: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

The Reachability Problem forSPDIs

e3

e10

e9e12

e11

e2

e4

e5

e8

e1

xf

x0e7e6

Reachability problem: Is there a trajectory from x0 to xf?

Algorithmic Analysis of Polygonal Hybrid Systems – p.11/66

Page 31: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

The Reachability Problem forSPDIs

e3

e10

e9e12

e11

e2

e4

e5

e8

e1

xf

x0e7e6

Reachability problem: Is there a trajectory from x0 to xf?

Algorithmic Analysis of Polygonal Hybrid Systems – p.11/66

Page 32: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Solving the Reachability Prob-lem

1. From trajectories to simplified trajectories

Algorithmic Analysis of Polygonal Hybrid Systems – p.12/66

Page 33: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

1. Simplification of trajectories

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

Theorem: If there is an arbitrary trajectory betweentwo points then it always exists a straightenednon–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

Page 34: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

1. Simplification of trajectories

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

Theorem: If there is an arbitrary trajectory betweentwo points then it always exists a straightenednon–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

Page 35: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

1. Simplification of trajectories

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

R8

x′

Theorem: If there is an arbitrary trajectory betweentwo points then it always exists a straightenednon–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

Page 36: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

1. Simplification of trajectories

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

R8

Theorem: If there is an arbitrary trajectory betweentwo points then it always exists a straightenednon–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

Page 37: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

1. Simplification of trajectories

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

Theorem: If there is an arbitrary trajectory betweentwo points then it always exists a straightenednon–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

Page 38: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

1. Simplification of trajectories

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

Theorem: If there is an arbitrary trajectory betweentwo points then it always exists a straightenednon–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

Page 39: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

1. Simplification of trajectories

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

Theorem: If there is an arbitrary trajectory betweentwo points then it always exists a straightenednon–crossing trajectory between them

Algorithmic Analysis of Polygonal Hybrid Systems – p.13/66

Page 40: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Solving the Reachability Prob-lem

1. From trajectories to simplified trajectories

2. From simplified trajectories to signatures

Algorithmic Analysis of Polygonal Hybrid Systems – p.14/66

Page 41: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

2. Abstraction into signatures

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

σ = e1e2e3 . . . e5e6e7 . . . e13e6e7e8e15

Algorithmic Analysis of Polygonal Hybrid Systems – p.15/66

Page 42: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Solving the Reachability Prob-lem

1. From trajectories to simplified trajectories

2. From simplified trajectories to signatures

3. From signatures to factorized signatures

Algorithmic Analysis of Polygonal Hybrid Systems – p.16/66

Page 43: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

3. Factorization of SignaturesFor σ = e1e2e3 . . . e5e6e7 . . . e13e6e7e8e15

e11

e10

e9 e8

e7

e6

e5

e3

e4

e13

e14e15

e2

e1

e12

x′

x

R3

R1

R2R4

R5

R6

R7

R8R9

R10

R12

R11

We obtain the representation:σ = e1e2e3 (e4e1e2e3)

2e5e6e7e8 (e9 · · · e13e6e7e8)2 e15

Algorithmic Analysis of Polygonal Hybrid Systems – p.17/66

Page 44: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

3. Canonical Factorization ofSignatures

Representation Theorem: Any edge signatureσ = e1, e2, . . . , en can be represented as

σ = r1(s1)k1r2(s2)

k2 . . . rn(sn)knrn+1

• Properties:• ri is a seq. of pairwise different edges;

• si is a simple cycle;

• ri and rj are disjoint

• si and sj are different

Proof based on topological properties of the plane

Algorithmic Analysis of Polygonal Hybrid Systems – p.18/66

Page 45: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

3. Canonical Factorization ofSignatures

Representation Theorem: Any edge signatureσ = e1, e2, . . . , en can be represented as

σ = r1(s1)k1r2(s2)

k2 . . . rn(sn)knrn+1

• Properties:• ri is a seq. of pairwise different edges;

• si is a simple cycle;

• ri and rj are disjoint

• si and sj are different

Proof based on topological properties of the plane

Algorithmic Analysis of Polygonal Hybrid Systems – p.18/66

Page 46: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Solving the Reachability Prob-lem

1. From trajectories to simplified trajectories

2. From simplified trajectories to signatures

3. From signatures to factorized signatures

4. From factorized signatures to types of signatures

Algorithmic Analysis of Polygonal Hybrid Systems – p.19/66

Page 47: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

4. Types of SignaturesAbstraction: Any edge signature

σ = r1(s1)k1r2(s2)

k2 . . . rn(sn)knrn+1

belongs to a type

type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1

Prop. The set of types of signatures is finite

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

Page 48: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

4. Types of SignaturesAbstraction: Any edge signature

σ = r1(s1)k1r2(s2)

k2 . . . rn(sn)knrn+1

belongs to a type

type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1

s1 s2 sn

rn rn+1r3r2r1

Prop. The set of types of signatures is finite

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

Page 49: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

4. Types of SignaturesAbstraction: Any edge signature

σ = r1(s1)k1r2(s2)

k2 . . . rn(sn)knrn+1

belongs to a type

type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1

In the previous example:

type(σ) =e1e2e3, e4e1e2e3, e5e6e7e8, e9 · · · e13e6e7e8, e15

Prop. The set of types of signatures is finite

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

Page 50: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

4. Types of SignaturesAbstraction: Any edge signature

σ = r1(s1)k1r2(s2)

k2 . . . rn(sn)knrn+1

belongs to a type

type(σ) = r1, s1, r2, s2, . . . rn, sn, rn+1

Prop. The set of types of signatures is finite

Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

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Solving the Reachability Prob-lem

1. From trajectories to simplified trajectories

2. From simplified trajectories to signatures

3. From signatures to factorized signatures

4. From factorized signatures to types of signatures

5. Analysis of each type of signature (computingsuccessors)

Algorithmic Analysis of Polygonal Hybrid Systems – p.21/66

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Computing Successors (for σ)One step (σ = e1e2)

e4

e5

e11

e12

e9

e10

e2e3

e1

e13

e8

e7e6

x

I2[a1x + b1, a1x + b1]

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

Page 53: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Computing Successors (for σ)Several steps (σ = e1e2e3)

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

e4

e5

e11

e12

e9

e10

e2e3

e1

e13

e8

e7e6

I3

x

I3 = Succσ(x) = [a2x + b2, a′2x + b′2] ∩ e3

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

Page 54: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Computing Successors (for σ)Several steps (σ = e1e2e3e4e5)

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � �

x

e5

e4 e1

e10

e9

e12

e11

e6 e7

e8

e13

e2e3

I5 = Succσ(x) = [a4x + b4, a′4x + b′4] ∩ e5

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

Page 55: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Computing Successors (for σ)One cycle (σ = s = e1e2 · · · e8e1)

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �x

e5

e4

e8

e11

e10

e9

e12e13

e7e6

e3 e2

e1

I9

I9 = Succσ(x) = [a8x + b8, a′8x + b′8] ∩ e1

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

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Computing Successors (for σ)One cycle (σ = s = e1e2 · · · e8e1)

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �x

e5

e4

e8

e11

e10

e9

e12e13

e7e6

e3 e2

e1

I∗

u∗l∗

l∗ = a1l∗ + b1 u∗ = a2u

∗ + b2

I∗ = Succ∗σ(x) = [l∗, u∗] ∩ e1

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

Page 57: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Computing Successors (for σ)σ = (s)∗e13 (s = e1e2 · · · e8e1)

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

e8

e13

e1

e2

e4

e6 e7

e11

e10

e9

e12

e5

xI ′

e3

One cycle iterated: solution of fixpoint equation(acceleration): I ′ = Succe8e13

◦ Succe1···e8◦ Succ

∗s(x)

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

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Computing Successors

Lemma: Successors have the form

Succσ(l, u) = [a1l + b1, a2u + b2] ∩ J if [l, u] ⊆ S

Lemma: Fixpoint equations

[a1l∗ + b1, a2u

∗ + b2] = [l∗, u∗]

can be explicitely solved (without iterating).We have that (I = [l, u]):

Succ∗σ(I) = [l∗, u∗] ∩ J

Algorithmic Analysis of Polygonal Hybrid Systems – p.23/66

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Reachability Algorithm

for each type of signature τ docheck whether Reachτ(x0, xf)

To test whether Reachτ (x0, xf) for

τ = r1(s1)∗ · · · (sn)

∗rn+1

Compute Succr

Accelerate (Succs)∗

Algorithmic Analysis of Polygonal Hybrid Systems – p.24/66

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Reachability: Main Result

• The capability of computing fixpoints for simplecycles (acceleration)

• The set of types of signatures is finite

Reachability is decidable for SPDI

Algorithmic Analysis of Polygonal Hybrid Systems – p.25/66

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SPeeDI: a Tool for SPDIs

Algorithmic Analysis of Polygonal Hybrid Systems – p.26/66

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Implementation: SPeeDI• We have implemented the reachability algorithm

for SPDIs: SPeeDI (joint work with GordonPace)

• Language: Haskell

<trace><type_of_signature> simsig2figsimsigreachable <file.fig>

NO

YES

<file.spdi>

<input interval>

<exit interval>

Algorithmic Analysis of Polygonal Hybrid Systems – p.27/66

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Implementation: SPeeDI• We have implemented the reachability algorithm

for SPDIs: SPeeDI (joint work with GordonPace)

• Language: Haskell

<trace><type_of_signature> simsig2figsimsigreachable <file.fig>

NO

YES

<file.spdi>

<input interval>

<exit interval>

Algorithmic Analysis of Polygonal Hybrid Systems – p.27/66

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Implementation: SPeeDI

Animate

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

Page 65: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Implementation: SPeeDI

Animate

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

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Implementation: SPeeDI

Animate

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

Page 67: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Implementation: SPeeDI

Animate

Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

Page 68: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Phase Portrait of SPDIs

Algorithmic Analysis of Polygonal Hybrid Systems – p.29/66

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Phase Portrait

Phase Portrait: a picture of important objects of adynamical system

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

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Phase Portrait

Phase Portrait: a picture of important objects of adynamical system

e3

e9e12

e2

e6 e7

e4

e5

e8

e1

e10e11

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

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Phase Portrait

Phase Portrait: a picture of important objects of adynamical system

e3

e9e12

e2

e6 e7

e4

e5

e8

e1

e10e11

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Page 72: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Phase Portrait

Phase Portrait: a picture of important objects of adynamical system

e3

e9e12

e2

e6 e7

e4

e5

e8

e1

e10e11

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Page 73: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Phase Portrait

Phase Portrait: a picture of important objects of adynamical system

e3

e9e12

e2

e6 e7

e4

e5

e8

e1

e10e11

Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Page 74: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Viability KernelViab(σ): Is the greatest set of initial points oftrajectories which can cycle forever in σ

Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

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Viability KernelViab(σ): Is the greatest set of initial points oftrajectories which can cycle forever in σ

Example: σ = e1e2 . . . e8e1

R6 R8

R4

R3

R7

R1R5

R2

e5

e4

e3e2

e1

e8

e7e6

Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

Page 76: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Viability KernelViab(σ): Is the greatest set of initial points oftrajectories which can cycle forever in σ

Example: σ = e1e2 . . . e8e1

R6 R8

R4

R3

R7

R1R5

R2

e5

e4

e3e2

e1

e8

e7e6

Theorem: Viab(σ) = Preσ(Dom(Succσ))

Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

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Controllability KernelCntr(σ): Is the greatest set of mutually reachablepoints via trajectories that remain in the cycle

Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

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Controllability KernelCntr(σ): Is the greatest set of mutually reachablepoints via trajectories that remain in the cycle

Example: σ = e1e2 . . . e8e1

R6 R8

R3

R7

R4 R2

R1R5

l u

e2e3

e1

e7e6

e4

e8e5

Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

Page 79: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Controllability KernelCntr(σ): Is the greatest set of mutually reachablepoints via trajectories that remain in the cycle

Example: σ = e1e2 . . . e8e1

R6 R8

R3

R7

R4 R2

R1R5

l u

e2e3

e1

e7e6

e4

e8e5

Theorem: Cntr(σ) = (Succσ ∩ Preσ)(CD(σ))

Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

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Viability Kernel

Algorithm: phase portrait for SPDIs

for each simple cycle σ doCompute Viab(σ) (viability kernel)Compute Cntr(σ) (controllability kernel)

Algorithmic Analysis of Polygonal Hybrid Systems – p.33/66

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Viability Kernel

Algorithm: phase portrait for SPDIs

for each simple cycle σ doCompute Viab(σ) (viability kernel)Compute Cntr(σ) (controllability kernel)

Both kernels are exactly computed by non-iterativealgorithms!

Algorithmic Analysis of Polygonal Hybrid Systems – p.33/66

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Properties of the KernelsTheorem: Any viable trajectory in σ converges toCntr(Kσ)

R6 R8

R4 R2

R3

R7

R5 R1

e6 e7

e8

e1

e2

e3

e4

e5

• Controllability Kernel: “Weak” analog of limit cycle

• Viability Kernel: Its “local” attraction basinAlgorithmic Analysis of Polygonal Hybrid Systems – p.34/66

Page 83: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Convergence PropertiesEvery trajectory with infinite signature withoutself-crossings converges to the controllabilitykernel of some simple edge-cycle

R5

R4

R3

R2

R11

R12

e12

R13

R14R15

R1

R8

R7

R6

e5

e4

e3e2

e1

e8

e7e6

e15

e14

e13

e11

e10

Algorithmic Analysis of Polygonal Hybrid Systems – p.35/66

Page 84: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Between Decidable andUndecidable

Algorithmic Analysis of Polygonal Hybrid Systems – p.36/66

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More complex 2-dim systemsWhat happens if . . .

• . . . we allow jumps?• . . . the PCD is on a 2-dim surface/manifold?• . . . ?

Answer: Reachability is equivalent to a well knownopen problem

Algorithmic Analysis of Polygonal Hybrid Systems – p.37/66

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More complex 2-dim systemsWhat happens if . . .

• . . . we allow jumps?• . . . the PCD is on a 2-dim surface/manifold?• . . . ?

Answer: Reachability is equivalent to a well knownopen problem

Algorithmic Analysis of Polygonal Hybrid Systems – p.37/66

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Our Reference Model1-dim Piecewise Affine Maps (PAMs):f : R → R, f(x) = aix + bi for x ∈ Ii

Reachability? Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

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Our Reference Model1-dim Piecewise Affine Maps (PAMs):f : R → R, f(x) = aix + bi for x ∈ Ii

I3

R

I5I2

a1x + b1

I4I1

Reachability? Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Page 89: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Our Reference Model1-dim Piecewise Affine Maps (PAMs):f : R → R, f(x) = aix + bi for x ∈ Ii

I3

R

I5I2

a1x + b1

a5x + b5

I4I1

Reachability? Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Page 90: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Our Reference Model1-dim Piecewise Affine Maps (PAMs):f : R → R, f(x) = aix + bi for x ∈ Ii

I3

R

I5I2

a4x + b4

a1x + b1

a5x + b5

I4I1

Reachability? Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Page 91: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Our Reference Model1-dim Piecewise Affine Maps (PAMs):f : R → R, f(x) = aix + bi for x ∈ Ii

I3

R

a2x + b2

I5I2

a4x + b4

a1x + b1

a5x + b5

I4I1

Reachability? Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Page 92: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Our Reference Model1-dim Piecewise Affine Maps (PAMs):f : R → R, f(x) = aix + bi for x ∈ Ii

I3

R

a2x + b2

I5I2

a4x + b4

a1x + b1

a5x + b5

I4I1

Reachability?

Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Page 93: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

Our Reference Model1-dim Piecewise Affine Maps (PAMs):f : R → R, f(x) = aix + bi for x ∈ Ii

I3

R

a2x + b2

I5I2

a4x + b4

a1x + b1

a5x + b5

I4I1

Reachability? Open problem!

Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Page 94: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

PCD on 2-dim manifolds (PCD2m)Example: Torus

R2

R1

R3 R4

Reachability?

Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

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PCD on 2-dim manifolds (PCD2m)Example: Torus

R2

R1

R3 R4

Reachability?

Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

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PCD on 2-dim manifolds (PCD2m)Example: Torus

R2

R1

R3 R4

Reachability?

Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

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PCD on 2-dim manifolds (PCD2m)Example: Torus

R2

R1

R3 R4

Reachability?

Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

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PCD on 2-dim manifolds (PCD2m)Example: Torus

R2

R1

R3 R4

Reachability?

Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

Page 99: Algorithmic Analysis of Polygonal Hybrid Systems - Chalmersgersch/slides-talks/slides-PhD-thesis.pdf · Algorithmic Analysis of Polygonal Hybrid Systems GERARDO SCHNEIDER VERIMAG

PCD on 2-dim manifolds (PCD2m)Example: Torus

R2

R1

R3 R4

Reachability?

Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

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PCD on 2-dim manifolds (PCD2m)Example: Torus

R2

R1

R3 R4

Reachability?

Theorem: PCD2m ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

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Hierarchical PCDs (HPCD)

x := ax + by := 0

R1

R4R3

(a1, b1)

(a4, b4)(a3, b3)

(a2, b2)

R2

PCD2

`3

`2`1

`2

PCD3

PCD1

g

g

x := ax + bg

y := 0

x := ax + by := 0

Reachability?

Theorem: HPCD ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.40/66

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Hierarchical PCDs (HPCD)

x := ax + by := 0

R1

R4R3

(a1, b1)

(a4, b4)(a3, b3)

(a2, b2)

R2

PCD2

`3

`2`1

`2

PCD3

PCD1

g

g

x := ax + bg

y := 0

x := ax + by := 0

Reachability?

Theorem: HPCD ≡ PAM

Algorithmic Analysis of Polygonal Hybrid Systems – p.40/66

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Undecidable 2-dim Systems

Algorithmic Analysis of Polygonal Hybrid Systems – p.41/66

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Undecidability Results• HPCDs with One Counter (HPCD1c)• HPCDs with Infinite Partition (HPCD∞)• Origin-dependent rate HPCDs (HPCDx)

Reachability? UNDECIDABLE!

Theorem:HPCD1c, HPCD∞ and HPCDx

simulateTuring machines

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Undecidability Results• HPCDs with One Counter (HPCD1c)• HPCDs with Infinite Partition (HPCD∞)• Origin-dependent rate HPCDs (HPCDx)

Reachability?

UNDECIDABLE!

Theorem:HPCD1c, HPCD∞ and HPCDx

simulateTuring machines

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Undecidability Results• HPCDs with One Counter (HPCD1c)• HPCDs with Infinite Partition (HPCD∞)• Origin-dependent rate HPCDs (HPCDx)

Reachability? UNDECIDABLE!

Theorem:HPCD1c, HPCD∞ and HPCDx

simulateTuring machines

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Summary of Results

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Summary of Results

PCD

A B "A is a particular case of B"

SPDI

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Summary of Results

PCD

A B "A is a particular case of B"

SPDI

decidable

Controllability kernel

Convergence propertiesViability kernel

Exact computationAbstractionAccelerationPoincare mapSPeeDI

Reachability analysisPhase Portrait

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Summary of Results

HPCD

PCD

A B "A is a particular case of B"

SPDI

decidable

Reachability analysisPhase Portrait

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Summary of Results

HPCD

PCD

A B "A is a particular case of B"

SPDI

decidableHPCDx

HPCD∞

HPCD1c

Reachability analysisPhase Portrait

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Summary of Results

TM HPCD

PCD

A

A B

B "A is a particular case of B"

"A is simulated by B"

SPDI

decidable

undecidable

HPCDx

HPCD∞

HPCD1c

Reachability analysisPhase Portrait

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Summary of Results

TM HPCD

PCD

A

A B

B "A is a particular case of B"

"A is simulated by B"

SPDI

decidable

undecidable

HPCDx

HPCD∞

HPCD1cRA2cl

PCD2m

HPCDiso

RA1sk1sl

LAstRA1cl1mc

Reachability analysisPhase Portrait

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Summary of Results

TM HPCD

PCD

A

A

PAM

B

B "A is a particular case of B"

"A is simulated by B"

SPDI

decidable

undecidabledo not know

HPCDx

HPCD∞

HPCD1c

PAMinj

RA2cl

PCD2m

HPCDiso

RA1sk1sl

LAstRA1cl1mc

Reachability analysisPhase Portrait

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Perspectives• SPDI to approximate non-linear differential

equations• Conditions for decidability of PCDs on 2-dim

manifolds• Application of the geometric method to higher

dimensions• Extension of SPeeDI: algorithm for viability and

controllability kernels• SPeeDI: “Topological” optimizations

Algorithmic Analysis of Polygonal Hybrid Systems – p.45/66

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Merci!

Gracias!

Obrigado!

(Brasil Penta-Campeão!)

Thank you!

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Theorem de Poincaré-Bendixson

A non-empty compact limit set of C1 planardynamical system that contains no equilibrium pointsis a close orbit.

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Comparison with HyTechExample:

(1,−2)

(−1,−2)

(−1, 9

10)

(1, 1)

(−1, 1

10)

Fixpoint: I∗ = (2009

; 200)

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Comparison with HyTech

Final Point HyTech SPeeDI Reachable

199 overflow 0.05 sec Yes

200 overflow 0.05 sec No

201 overflow 0.01 sec No

210 overflow 0.05 sec No

5 0.04 sec 0.05 sec No

20 0.07 sec 0.05 sec No

2009

0.10 sec 0.05 sec Yes

2019

overflow 0.03 sec Yes

1999

0.07 sec 0.04 sec Yes

12

0.06 sec 0.05 sec No

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Comparison with HyTechSimulation of reachability for xf = 201

9

l1 =203

10l∗ =

200

9If =

201

9u1 =

118

5

3 4

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e6 e7

e2e3

e4

e5

R6

R7

R8

R1

R2

R3

R4

R5

e8

e1

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Composition of TAMFsTAMFs are closed under composition:For

F1(x) = F1({x} ∩ S1) ∩ J1

andF2(x) = F2({x} ∩ S2) ∩ J2

we have that

F2 ◦ F1(x) = FF ′,S′,J ′(x)

withF ′ = F2 ◦ F1,J ′ = J2 ∩ F2(J1 ∩ S2) andS ′ = S1 ∩ F−1

1 (J1 ∩ S2)

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Reachability Algorithm (Exam-ple)

e5

e4

e3

e2

e1

e8

e7e6

R1

R5

R4

R8R6

x0 = 1

2

R2R3

R7

xf = 4

5

• Type of signature: σ = (e1 · · · e8)∗

• Successor for the loop s = e1 . . . e8:

Succs(l, u) = [ l2− 1

20, u

2+ 23

60] ∩ (1

5, 1)

if [l, u] ⊆ (0, 1)Algorithmic Analysis of Polygonal Hybrid Systems – p.51/66

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Reachability Algorithm (Exam-ple)

• Fixpoint equation: Succe1...e8(I∗) = I∗

• Solution: I∗ = [l∗, u∗] = [15, 23

30]

• Hence: Succe1...e8(x0) ⊆ [1

5, 23

30]

...e4

e3

e2

e6

e1

e8

e7

e5

u∗ = 23

30

xf = 4

51

5

x0 = 1

2

Conclusion: xf 6∈ [15, 23

30]).

Hence, xf 6∈ Reach(x0)Note: the solution was found without iterating (byacceleration).

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Viability KernelK

A

Viab(K) = A ∪ B

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Viability KernelK

A

Viab(K) = A ∪ B

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Viability KernelK

A

Viab(K) = A ∪ B

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Viability KernelK

BA w

x

z

Viab(K) = A ∪ B

• M is a viability domain if ∀x ∈ M , ∃ at least onetrajectory ξ, starting in x and remaining in M

• Viab(K): Viability kernel of K is the largestviability domain M contained in K

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Viability Kernel for SPDIs• We can easily compute the Viability Kernel for

one cycle, which is a polygon

• Theorem: Viab(Kσ) = Preσ(Dom(Succσ))

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Viability Kernel for SPDIs• We can easily compute the Viability Kernel for

one cycle, which is a polygon

e6 e7

e2e3

e4

e5

R6

R7

R8

R1

R2

R3

R4

R5

e8

e1

• Theorem: Viab(Kσ) = Preσ(Dom(Succσ))

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Viability Kernel for SPDIs• We can easily compute the Viability Kernel for

one cycle, which is a polygon

R6 R8

R4

R3

R7

R1R5

R2

e5

e4

e3e2

e1

e8

e7e6

• Theorem: Viab(Kσ) = Preσ(Dom(Succσ))

Algorithmic Analysis of Polygonal Hybrid Systems – p.54/66

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Controllability Kernel

A

K

Cntr(K) = A

• M is controllable if ∀x,y ∈ M , ∃ a trajectorysegment ξ starting in x that reaches an arbitrarilysmall neighborhood of y without leaving M

• Controllability kernel of K, denoted Cntr(K), isthe largest controllable subset of K

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Controllability Kernel

A

K

Cntr(K) = A

• M is controllable if ∀x,y ∈ M , ∃ a trajectorysegment ξ starting in x that reaches an arbitrarilysmall neighborhood of y without leaving M

• Controllability kernel of K, denoted Cntr(K), isthe largest controllable subset of K

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Controllability Kernel

A

K

Cntr(K) = A

• M is controllable if ∀x,y ∈ M , ∃ a trajectorysegment ξ starting in x that reaches an arbitrarilysmall neighborhood of y without leaving M

• Controllability kernel of K, denoted Cntr(K), isthe largest controllable subset of K

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Controllability Kernel

A

K

x

w

z

Cntr(K) = A

• M is controllable if ∀x,y ∈ M , ∃ a trajectorysegment ξ starting in x that reaches an arbitrarilysmall neighborhood of y without leaving M

• Controllability kernel of K, denoted Cntr(K), isthe largest controllable subset of K

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Controllability Kernel for SPDIs

e6 e7

e2e3

e4

e5

R6

R7

R8

R1

R2

R3

R4

R5

e8

e1

• Theorem: Cntr(Kσ) = (Succσ ∩ Preσ)(CD(σ))(We know how to compute the special intervalCD(σ) = [l, u])

Algorithmic Analysis of Polygonal Hybrid Systems – p.56/66

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Controllability Kernel for SPDIs

R6 R8

R3

R7

R4 R2

R1R5

l u

e2e3

e1

e7e6

e4

e8e5

• Theorem: Cntr(Kσ) = (Succσ ∩ Preσ)(CD(σ))(We know how to compute the special intervalCD(σ) = [l, u])

Algorithmic Analysis of Polygonal Hybrid Systems – p.56/66

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PAM simulate HPCD

e1

e5

e2e0

e4e3

I1

I2

I3

x′ = a3x + b3

x

e1 e2 e3 e4I1 I2 I3

A2x + B2

A3x + B3

A4x + B4

e0

R

Algorithmic Analysis of Polygonal Hybrid Systems – p.57/66

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HPCD simulate PAM

e′

e

γ(e′, x, y) = (e, aix + bi, 0)

Ii

Algorithmic Analysis of Polygonal Hybrid Systems – p.58/66

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RA1cl1mc equivalent to PAM

x = 0

y = 1

0 ≤ y ≤ 1

y = 1 ∧ x ∈ Ii

x := aix + bi; y := 0

Algorithmic Analysis of Polygonal Hybrid Systems – p.59/66

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RA2cl equivalent to PAM

x = 1

y = 1

0 ≤ y ≤ 1

y = 1 ∧ x − 1 ∈ Ii

x := ai(x − 1) + bi; y := 0

e′

e

γ(e′, x, y) = (e, ai(x − 1) + bi, 0)

Ii + 1

Algorithmic Analysis of Polygonal Hybrid Systems – p.60/66

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RA1sk1sl equivalent to PAM

(a)

(b)

aj(uj − lj)ai(ui − li)

e′i

ei

(−1, ai)

ej

(1, aj)

e′j

PCDj PCDi

ujlj li ui

x = li

y := 0, x := y + fi(li)x = 1y = aj

0 ≤ y ≤ aj(uj − lj)

lj ≤ x ≤ uj

0 ≤ y ≤ ai(ui − li)

x = −1y = ai

li ≤ x ≤ ui

γ(e′

i, x, y) = (ej , y + fi(li), 0)

`j`i

Algorithmic Analysis of Polygonal Hybrid Systems – p.61/66

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From RA1sk1sl to LAst

Ci ≤ y ≤ Di Cj ≤ y ≤ Dj

x = 1y = ai

x = 1y = aj

Aj ≤ x ≤ BjAi ≤ x ≤ Bi

x = 0y = 1

where bi = Ci + Biai and bj = Cj + Bjaj

x = 1y = 0

x = 0y = 1

x = 1y = 0

y = −aix + bi y = −ajx + bj

y ≥ −aix + biCi ≤ y ≤ Di

x ≤ Bi

y ≤ −aix + bi

y ≥ Ci

Ai ≤ x ≤ Biy ≤ −ajx + bjAj ≤ x ≤ Bj

y ≥ Cj

y ≥ −ajx + bjCj ≤ y ≤ Dj

x ≤ Bj

y := Cj , x := y + di

x = Bi

y := Cj , x := y + di

x = Bi

Algorithmic Analysis of Polygonal Hybrid Systems – p.62/66

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PCD2m simulate PAMinj

(a) (b) (c)

aix − aili

Iji

Ili

Iki

Iih

Iim

aix + bi

v

e′

eli ui

v′

f(li)

−f(li)

Ii

Iji

Ili

Iki

Ij

Ik

Il

Algorithmic Analysis of Polygonal Hybrid Systems – p.63/66

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HPCD1c simulate TM

R2

e1

e2

R4R3

R1

R2R1

R4R3

e4

e3 e1e5

e6

e7

R6

R5

e2

v1v2

e3

e4

e8

PCDi PCD′i; PCD′′

i

Algorithmic Analysis of Polygonal Hybrid Systems – p.64/66

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HPCD1c simulate TMTM-state qi:

PCDi

PCD′i

PCD′′i

γ(e1, λ, c) = (e2, λ + 1, c − 1)

g ≡ e1 ∧ c > 0

γ(e1, λ, c) = (e2, λ + 1, c − 1)

g ≡ e1 ∧ c > 0

γ(e2, λ, c) = (e2, λ, c)

g ≡ e2 PCDj

PCDk

γ(e1, λ, c) = (e4, f ′′(λ), c)

g ≡ e1 ∧ c = 0

g ≡ e1 ∧ c = 0

g ≡ e1

γ(e1, λ, c) = (e4, λ − 1, c + 1)

γ(e3, λ, c) = (e2, λ, c)

g ≡ e3

qi

`i

`′′i

`′i `j

`k

γ(e1, λ, c) = (e4, f ′(λ), c)

Algorithmic Analysis of Polygonal Hybrid Systems – p.65/66

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HPCDx simulate TM

e2

e3

e1

(0, f(x0) = (−1)b2x0c)

(0, f(x0) = (−1)b2x0c)

f(x) =

{

1 if fracx < 12

−1 otherwise

Algorithmic Analysis of Polygonal Hybrid Systems – p.66/66